Sheryl Sandberg’s polarizing manifesto about the role of women in the workplace has drawn its share of praise and criticism.

But at Stanford University on Tuesday night, a new generation of engineers, entrepreneurs and future feminists enthusiastically embraced Facebook’s chief operating officer’s message of leadership.

Stirring the crowd to its feet at the sold-out lecture in support of her new book and burgeoning feminist movement dubbed “Lean In,” she asked, “How many of you want to be No. 1 in your field? What would you do if you were not afraid? Stand up if you can think of one thing you can do differently not just for yourself, but for women all over the world.”

Many found it inspiring.

“She can reshape corporate culture and national culture,” said Michele Elam, professor of English and director of curriculum at Stanford. “There are a lot of people doing work on gender scholarship, but it isn’t necessarily translatable, it hasn’t necessarily hit the public. She has made it her own and communicated it.”

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In a warm and disarming tone, Sandberg told humbling tales about crying at work and guilt about mothering.

Sandberg -- whose net worth was pegged at $500 million by Forbes last October -- described how a bossy little girl reached top management, steering a social network to once-unimaginable heights.

Her book, “Lean In,” has roared into the No. 1 position on the New York Times hardcover nonfiction list. Her speeches have gone viral, with some included in syllabuses at the Stanford and Harvard business schools. Her TED talk has been viewed more than 2 million times. Other women in the valley have been successful in tech, but none has been so vocal about female empowerment.

And it struck a chord.

“I could completely relate -- especially when young women who speak up are told they are bossy,” said Lina Malova, 28, a Kenyan native who is a policy fellow with the San Francisco Department on the Status of Women. “Letting the fear go -- that’s what I’ve learned,” she said. “When I am at the table, it is my place. I just have to be there.”

As for Facebook, Sandberg said it has a female head of global sales, head of design and head of policy. “The most important thing we do differently at Facebook,” she said, “is that we talk about these issues.”

The lecture, sponsored by Stanford’s Clayman Institute for Gender Research, started out with some hard truths.

Men still run the world, she said, noting that of 197 heads of state, only 22 are women; of the top 500 companies by revenues, only 21 are headed by women; and in politics, women hold just 18 percent of congressional offices. In 2010, women earned just 77 cents for every dollar men made.

She offered this advice:

About progressing on the job: “I’m a pragmatist. I think, as a woman, you have to be more careful. You have to be more communal, you have to say yes to more things than men, you have to worry about things that men don’t have to worry about. But once we get enough women into leadership, we can break stereotypes down. If you lead, you get to decide.”

Dating: “The most important life decision is: Who your partner is. That decision will determine more of your career than anything else you do. There are no senior women leaders who don’t have a really supportive partner at home. Without it, you won’t achieve what you want to achieve.”

Marissa Mayer’s decision to end telecommuting: “It is really hard to know what is happening at Yahoo. There have been no public statements. But the scrutiny, the media firestorm, is about her being a woman. Full stop. Best Buy did the same thing. When it’s a man, no one pays attention.”

Changing public policy: “I am all for public policy and institutional reform. I also believe that it is not enough.

“Progress will turn more on private conversations, not public conversations,” she said. “The conversations with our workers and our supervisors and our partners and children -- they are not always fun. But if they happen, we will get to a better world.”